thumb|380px|A globular set with 0-cells (vertices), 1-cells (gray edges), 2-cells (red edges), and 3-cells (blue edges). The source and target of each <math>k</math>-cell must be single (<math>k</math>-1)-cells. For example, the red edge ''A'' connects single 1-cells ''a'' and ''b'', while ''B'' connects ''c'' and ''d'', and ''C'' forms a self-connection on ''c''.
In category theory, a branch of mathematics, a '''globular set''' is a higher-dimensional generalization of a directed graph. Precisely, it is a sequence of sets <math>X_0, X_1, X_2, \dots</math> equipped with pairs of functions <math>s_n, t_n: X_n \to X_{n-1}</math> such that * <math>s_n \circ s_{n+1} = s_n \circ t_{n+1},</math> * <math>t_n \circ s_{n+1} = t_n \circ t_{n+1}.</math> (Equivalently, it is a presheaf on the category of “globes”.) The letters "''s''", "''t''" stand for "source" and "target" and one imagines <math>X_n</math> consists of directed edges at level ''n''.
In the context of a graph, each dimension is represented as a set of '''<math>k</math>-cells'''. Vertices would make up the 0-cells, edges connecting vertices would be 1-cells, and then each dimension higher connects groups of the dimension beneath it.<ref name="nlab_c"/><ref name="nlab_g"/>
It can be viewed as a specific instance of the polygraph. In a polygraph, a source or target of a <math>k</math>-cell may consist of an entire path of elements of (<math>k</math>-1)-cells, but a globular set restricts this to singular elements of (<math>k</math>-1)-cells.<ref name="nlab_c">{{nlab|id=computad}}</ref><ref name="nlab_g">{{nlab|id=globular+set}}</ref>
A variant of the notion was used by Grothendieck to introduce the notion of an ∞-groupoid. Extending Grothendieck's work,<ref>{{cite arXiv |last=Maltsiniotis |first=G |author-link= |title=Grothendieck ∞-groupoids and still another definition of ∞-categories |class=18C10, 18D05, 18G55, 55P15, 55Q05 |date=13 September 2010 |eprint=1009.2331 }}</ref> gave a definition of a weak ∞-category in terms of globular sets.
== References == {{reflist}} == Further reading == *Dimitri Ara. On the homotopy theory of Grothendieck ∞ -groupoids. ''J. Pure Appl. Algebra'', 217(7):1237–1278, 2013, arXiv:1206.2941 .
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Category:Category theory